1. Field of the Invention
Embodiments of the present invention relate to systems and methods for compressive sampling in spectroscopy and spectral imaging. More particularly, embodiments of the present invention relate to systems and methods for using less than one measurement per estimated signal value in spectroscopy and spectral imaging.
2. Background Information
A signal is a physical phenomenon distributed over a space. Examples include signals distributed over time, such as electromagnetic waves on antennas or transmission lines, signals distributed over Fourier space, such as optical or electrical spectra, and multidimensional signals distributed over physical space, such as 2D and 3D images. Digital signal analysis consists of signal estimation from discrete measurements. For many years, sampling theory formed the theoretical core of signal analysis. Basic signal characterization required regular sampling at a rate proportional to the signal bandwidth. The minimal sampling rate is termed the Nyquist frequency.
Over the past couple of decades, however, novel sampling approaches have emerged. One generalized sampling approach envisioned a bank of filtered copies of a signal at sub-Nyquist rates. Since this generalized sampling approach, a number of generalized sampling strategies and analyses have emerged. Particular attention has been given sampling on multiresolution bases and to irregular sampling. Multiresolution sampling is regarded as a generalization of generalized sampling and is used as a means of balancing sampling rates and parallel signal analysis.
Signal compression technology has developed in parallel with generalized sampling theory. Multiresolution representations, implemented using fractal and wavelet analysis, have been found to be critically enabling for signal compression on the basis of the empirically observed self-similarity of natural signals on multiple scales. For appropriate bases, natural signals yield sparse multiscale representations. Sparsity and heirarchical self-similarity have been combined in signal compression algorithms such as the embedded zero-tree and set partitioning in hierarchical trees algorithms.
Generalized sampling and compression technologies resolve specific challenges in signal processing systems. Generalized sampling enables systems to sample at lower rates than the Nyquist frequency, and data compression enables post-sampling reduction of the system data load.
Previous examples of sub-Nyquist sampling are divided into parameter estimation approaches and interpolation approaches. Parameter estimation uses sampled data to fit an a priori signal model. The signal model typically involves much greater constraints than conventional band limits. For example, one can assume that the signal consists of a single point source or a source state described by just a few parameters. As an example of parameter estimation, several studies have considered estimation of the frequency or position of an unknown but narrow-band source from sub-Nyquist samples. Interpolation generates new samples from measured data by curve fitting between known points.
In view of the foregoing, it can be appreciated that a substantial need exists for systems and methods that can advantageously provide for sub-Nyquist sampling that can be used in spectroscopy and spectral imaging.